from Dmitri Tymoczko

I'd also like to make a few remarks concerning James Tenney's graph,
which Daniel Wolf posted on the web.

1) If we think of this space as continuous, then the X axis rather
hard to interpret -- moving left or right doesn't change the pitch
that's being represented in any way. So for some (but not all)
musical purposes, the line at the center encodes all the needed
information. Neither Tenney nor I invented this line, which is just
log-frequency space.
Typically, a (continuous) n-dimensional tuning lattice can be
written as the product of a single line that represents log-
frequency space and an (n-1) dimensional space whose meaning is a
hard to interpret in the above sense. My sense is that this
redundancy often arises only when you try to make the tuning
lattices continuous -- for instance in a discrete 2 x 3 tuning grid,
every vertex corresponds to a distinct pitch, because the generators
are relatively prime.

2) As Daniel Wolf observes, any chord of pitches can be represented
as a set of points on the central Y axis of Tenney's graph, and that
voice leading distances can be represented by how far the points
have to move to get from one configuration to another. This is just
to say that we can represent chords of pitches as unordered
collections of points in log-frequency space, and that voice leading
distance is represented by the distance between one configuration
and another.
The problem is that this representation doesn't perspicuously
capture the natural music-theoretical idea that chords containing
the pitch classes are similar objects: for instance {C4, E4, G4} and
{C4, G4, E5} are represented by distinct configurations of points on
the central axis, whereas -- for many (but not all) music-
theoretical purposes -- it is useful to think of them as instances
of the same object, the C major chord.

3) To this end, it is useful to represent chords as sets of
unordered points on a circle, so that the C major chord is {C, E,
G}, or {12, 4, 7} on the circumference of an ordinary clock face.
The problem with this representation is that it doesn't clearly
depict the relation between a chord's internal structure and it's
voice leading possibilities.
(Again, one could "Tenney-ify" the circle by adding additional
dimensions, and then one could use these dimensions to draw tuning
lattices of various sorts. From the standpoint of voice leading we
don't need to do this, but for thinking about tuning it might well
be useful.)

4) Given 3), it's useful to look at the configuration space of
unordered points on a circle -- i.e. the geometrical spaces in which
each point represents a *configuration* of points on the circle.
These are the orbifolds I discuss in my paper. In these
configuration spaces a point represents a chord, and a line segment
represents a voice leading between two chords.
Happily, when you understand these orbifolds, the connection between
a chord's internal structure and its voice leading capabilities
becomes perfectly clear. The musical properties of "evenness"
and "unevenness" simply correspond to the distance from the center
of the orbifold; transpositional symmetry is represented by one set
of discrete symmetries of the orbifold; and inversional symmetry is
represented by another set of reflectional symmetries of the
orbifold.

I don't want to make any grand claims for my work. In some sense,
it's quite surprising that these spaces weren't discovered 100 years
ago. But I do think that this is relatively new territory, and
rather different from the discrete lattices familiar from tuning
theory.

DT
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Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri